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**On the asymptotic behavior of the solutions of third order delay differential equations.**
*(English)*
Zbl 1321.34097

The authors consider the nonlinear third order differential equation with constant delay
\[
[\Psi (x){x}'{]}'' + a(t){x}'' + b(t)\Phi (x){x}' + c(t)f(x(t - r)) = e(t), \tag{1}
\]
where \(r \in \mathbb R , \mathbb R = ( - \infty ,\infty ),r > 0\) is the constant delay, \(t \in \mathbb R ^{+}, \mathbb R ^ + = [0,\infty ), \quad a(t),b(t),c(t),e(t),\Psi (x),\Phi (x)\) and \(_{ }f(x)\) are continuous functions, and the derivatives \({\Psi }'(x),{\Phi }'(x)\) and \(_{ }{f}'(x)\) exist and are continuous for all \(x \in\mathbb R\).

The authors give sufficient conditions for the asymptotically stability and boundedness of the solutions of equation ({1}), when \(e(t) = 0\) and \(e(t) \neq 0\), respectively. By defining a suitable Lyapunov functional, they prove two new theorems on the subject. The obtained results extend the results in the literature. They also give an example for the illustrations.

The authors give sufficient conditions for the asymptotically stability and boundedness of the solutions of equation ({1}), when \(e(t) = 0\) and \(e(t) \neq 0\), respectively. By defining a suitable Lyapunov functional, they prove two new theorems on the subject. The obtained results extend the results in the literature. They also give an example for the illustrations.

Reviewer: Cemil Tunç (Van)

### MSC:

34K25 | Asymptotic theory of functional-differential equations |

34K20 | Stability theory of functional-differential equations |

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\textit{M. Remili} and \textit{D. Beldjerd}, Rend. Circ. Mat. Palermo (2) 63, No. 3, 447--455 (2014; Zbl 1321.34097)

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### References:

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